This section is intended to introduce various aspects of the art, which may be associated with exemplary embodiments of the present techniques. This discussion is believed to assist in providing a framework to facilitate a better understanding of particular aspects of the present techniques. Accordingly, it should be understood that this section should be read in this light, and not necessarily as admissions of prior art.
Hydrocarbons are widely used for fuels and chemical feedstocks. Hydrocarbons are generally found in subsurface rock formations that can be termed “reservoirs.” Removing hydrocarbons from the reservoirs depends on numerous physical properties of the rock formations, such as the permeability of the rock containing the hydrocarbons, the ability of the hydrocarbons to flow through the rock formations, and the proportion of hydrocarbons present, among others.
Often, mathematical models termed “simulation models” are used to simulate hydrocarbon reservoirs and optimize the production of the hydrocarbons. The goal of a simulation model is generally to simulate the flow patterns of the underlying geology in order to optimize the production of hydrocarbons from a set of wells and surface facilities.
The simulation model is a type of computational fluid dynamics simulation where a set of partial differential equations (PDE's) which govern multi-phase, multi-component fluid flow through porous media and the connected facility network is approximated and solved.
The set of governing differential equations vary with the physical process being modeled in the reservoir. They usually include volume balance, mass conservation, energy conservation, among others. The simulation is an iterative, time-stepping process where a particular hydrocarbon production strategy is optimized.
To perform fast and accurate reservoir simulation, discretization and formulation methods are of central importance because they deal with how mathematical equations governing multiphase flow are solved. Simulation models discretize the underlying PDEs on a structured (or unstructured) grid, which represents the reservoir rock, wells, and surface facility network. State variables, such as pressure and saturation, are defined at each grid block.
The discretization step casts the partial differential equations into discrete form suitable for computers to solve. The formulation method determines the overall solution procedure. The formulation method also determines how different equations and unknowns interact in the process. Unknowns may include unknown variables, such as pressure, saturation, component masses, etc.
Discretization methods include finite difference/finite volume, finite element, mixed finite element, discontinuous Galerkin, mimetic finite difference, etc. Today, two-point finite volume schemes are commonly used because they are easy to implement and fast to execute. However, two-point finite volume schemes have been found to compare unfavorably in terms of accuracy against multipoint finite volume methods or finite element type schemes. See Aavatsmark, I., “An Introduction to Multipoint Flux Approximation for Quadrilateral Grids,” Computational Geosciences, 6, 405-432 (2002); see also Chen, Q., Wan, J., Yang, Y. and Mifflin, R., “A New Multipoint Flux Approximation for Reservoir Simulation,” paper SPE 106464 presented at the 2007 SPE Symposium on Reservoir Simulation, The Woodlands, Tex., Feb. 26-27, 2007; see also Hoteit, H. and Firoozabadi, A., “Compositional Modeling by the Combined Discontinuous Galerkin and Mixed Methods,” SPE J., 19-34. (2006); see also Coats, K. H., “A Note on IMPES and Some IMPES-Based Simulation Models,” SPE J., 248 (2000).
Formulation methods include IMPES, Sequential Implicit (SI), IMPSAT, and Coupled Implicit (CI). All these methods have been used extensively in solving practical problems in simulation models.
The different formulation methods produce very different results in terms of stability, accuracy, and efficiency. With CI, all primary unknown variables which appear in the volume and mass conservation equations are solved implicitly. See Blair, P. M. and Weinaug, C. F., “Solution of Two-Phase Flow Problems Using Implicit Difference Equations,” Trans. SPE of AIME, 246, 417-424 (1969); see also Coats, K. H., “An Equation of State Compositional Model,” SPE J., 363-376 (1980). It is well-known that CI takes more time to solve for each time step, but produces stable simulation results even for large time step sizes.
Because of its stability, CI has been particularly effective in modeling challenging flow behaviors, for example, thermal process or water/gas coning near wells. See Mifflin, R. T. and Watts, J. W., “A Fully Coupled, Fully Implicit Reservoir Simulator for Thermal and Other Complex Reservoir Processes,” paper SPE 21252 presented at the 11th SPE Symposium on Reservoir Simulation, Anaheim, Calif., Feb. 17-20, 1991. Unfortunately, the cost for solving a CI system increases rapidly with the number of unknowns, which makes the technique impractical for modeling large reservoirs, especially when compositional models are used.
IMPES is at another end of the spectrum from CI. See Sheldon, J. W., Zondek, B., and Cardwell, W. T., “One-dimensional Incompressible, Non-Capillary, Two-Phase Fluid Flow in a Porous Medium,” Trans. SPE of AIME, 216, 290-296 (1959); see also Stone, H. L. and Garder, A. O. Jr., “Analysis of Gas-Cap or Dissolved-Gas Reservoirs,” Trans. SPE of AIME, 222, 92-104 (1961). In terms of implicitness, with the IMPES method, pressure is the only unknown variable solved implicitly. Saturation and component masses are updated explicitly after the pressure solve.
IMPES has been known to be fast per time step but tends to produce unstable results. For reservoirs which exhibit complex phase behaviors or significant pressure or saturation changes, numerical instability associated with IMPES could force simulation to take very small time steps, or cause it to bog down altogether before reaching the simulation end time.
Sequential Implicit (SI) and IMPSAT have been used to reach a desirable balance between stability and efficiency and they lie between IMPES and CI in terms of cost and stability. See MacDonald, R. C. and Coats, K. H., “Methods for Numerical Simulation of Water and Gas Coning,” Trans. SPE of AIME, 249, 425-43622 (1970); see also Spillette, A. G., Hillestad, J. G., and Stone, H. L., “A High-Stability Sequential Solution Approach to Reservoir Simulation,” SPE 4542, 48th Annual Fall Meeting, Las Vegas, Sep. 30-Oct. 3, (1973); see also Watts, J. W., “A Compositional Formulation of the Pressure and Saturation Equations,” SPE Reservoir Engineering, May, 243-252 (1986); see also Quandalle, P. and Savary, D., “An Implicit in Pressure and Saturations Approach to Fully Compositional Simulation,” paper SPE 18423 presented at the 10th SPE Symposium on Reservoir Simulation, Houston, Tex., Feb. 6-8, 1989; see also Branco, C. M. and Rodriguez, F., “A Semi-Implicit Formulation for Compositional Reservoir Simulation,” paper SPE 27053, SPE Advanced Technology Series, Vol. 4, No. 1, (1995); see also Cao, H. and Aziz K., “Performance of IMPSAT and IMPSAT-AIM Models in Compositional Simulation,” paper SPE 77720 presented at the SPE Annual Technical Conference and Exhibition held in San Antonio, Tex., Sep. 29-Oct. 2, 2002. Though SI was originally proposed to work with finite difference/volume method for modeling flows in reservoirs, researchers have recently used it with new discretization methods, including mixed finite element, multiscale control volume method, among others. See Arbogast, T., “Implementation of a Locally Conservative Numerical Subgrid Upscaling Scheme for Two-Phase Darcy Flow, Computational Geosciences,” 6, 453-481 (2002); see also Tchelepi, H. A., Jenny, P., Lee, S. H., and Wolfsteiner, C., “An Adaptive Multiphase Multiscale Finite Volume Simulator for Heterogeneous Reservoirs,” SPE 93395 presented at the 19th SPE Reservoir Simulation Symposium, The Woodlands, Tex., Jan. 31-Feb. 2, 2005.
With SI, each simulation time step consists of two solves. The first solves pressure unknowns at explicit saturation values, similar to IMPES. After the pressure solve, total flow velocity is computed. Flow rates are then expressed in terms of saturation unknowns, which are solved in the second step.
Because SI solves only one or two unknowns for each reservoir block in one time step, the computational cost for SI is very modest compared to CI. This is especially true for simulations of compositional models.
With SI, stability is increased due to the saturation step (the second step). While stability for SI is an improvement over IMPES, SI is still not as stable as CI.
The increased stability allows sequential runs to move forward when encountering difficult flow behaviors which would have triggered severe oscillations for IMPES. For this reason SI has been a practical option for the industry.
As the industry faces increasingly challenging production environments, however, engineers are building more and more complex reservoir models. It has been observed that with difficult simulation models, numerical instability from SI could produce oscillations in results severe enough to mask the true trend of flow responses in the reservoir. This behavior could have many undesirable consequences, for example, it could generate misleading data on gas-oil ratio, water cut and other parameters necessary for engineers to make well management decisions.
As a possible alternative to SI, IMPSAT solves a similar set of equations but is even closer to CI in terms of implicitness. For each time step, IMPSAT treats pressure and saturations as unknown variables and solves them simultaneously. Because of this, IMPSAT is more expensive than SI, but improves stability in situations where pressure and saturation behaviors are tightly coupled.
In general, the stability in formulation methods such as IMPES, SI, IMPSAT, and CI improves as the implicitness of the formulation method increases. Unfortunately, with the increase in implicitness, the cost of deriving solutions rises dramatically.
One approach to combat the implicitness versus cost balance combines formulation methods. By combining formulation methods, the adaptive implicit method (AIM), aims to take advantage of the strengths and avoid the weakness of the different formulation methods. In AIM, a reservoir may be dynamically divided into different regions based on certain stability criteria. The different formulation methods may then be targeted to the different regions. However, formulating robust stability criteria and creating an adaptable, highly efficient computational infrastructure are challenging tasks for AIM. Still, AIM has shown promising results. See Thomas, G. W. and Thurnau, D. H., “Reservoir Simulation Using an Adaptive Implicit Method,” SPE J, 959-968 (1983); see also Young, L. C., “Implementation of an Adaptive Implicit Method,” paper SPE 25245 presented at the 12th SPE Symposium on Reservoir Simulation held in New Orleans, La., Feb. 28-Mar. 3, 1993; see also Wan, J., Sarma, P., Usadi, A. K., and Beckner, B. L., “General Stability Criteria for Compositional and Black-Oil Models,” paper SPE 93096 presented at the SPE Symposium on Reservoir Simulation held in Houston, Tex., Jan. 31-Feb. 2, 2005; see also Lu, P., Shaw, J. S., Eccles, T. K., Mishev, I., and Beckner, B. L., “Experience With Numerical Stability, Formulation and Parallel Efficiency of Adaptive Implicit Methods,” paper SPE 118977 presented at the SPE Symposium on Reservoir Simulation held in Houston, Tex., Feb. 2-4, 2009.
There have been a number of research articles published on the general subject of formulation methods and partial differential equations for transport flow modeling. See, for example, Chen, Z. and Ewing, R., “Comparison of Various Formulations of Three-Phase Flow in Porous Media,” J. Comput. Phys., 1997, 132, 362-373; see also, Coats, K. H., George, W. D., and Marcum, B. E., “Three-Dimensional Simulation of Steamflooding,” Trans. SPE of AIME, 1974, 257, 573-592; see also, Coats, K. H., “Simulation of Steamflooding with Distillation and Solution Gas,” Soc. Petrol. Eng. J., 1976, 16, 235-247; see also Markovinovic, R. and Jansen, J. D., “Accelerating Iterative Solution Methods Using Reduced-Order Models as Solution Predictors,” Int. J. Numer. Meth. Engng., 2006, 68, 525-541; see also Trangenstein, J. A. and Bell, J. B., “Mathematical Structure of the Black-Oil Model for Petroleum Reservoir Simulation,” SIAM J. Applied Math., 1989, 49, 749-783; see also Trangenstein, J. A. and Bell, J. B., “Mathematical Structure of Compositional Reservoir Simulation,” SIAM J. Sci. Stat. Comput., 1989, 10, 817-845; see also Wong, T. W., Firoozabadi, A., Nutakki, R., and Aziz, K., “A Comparison of Two Approaches to Compositional and Black Oil Simulation,” paper SPE 15999 presented at the 9th SPE Symposium on Reservoir Simulation, San Antonio, Tex., Feb. 1-4, 1987; see also Peaceman, D., “Fundamentals of Numerical Reservoir Simulation,” Elsevier, Amsterdam-Oxford-New York, 1977; see also Lax, P. D. and Wendroff, B., “Systems of Conservation Laws,” Comm. Pure Appl. Math., 1960, 13, 217-237; see also Lax, P. D. and Wendroff, B., “Difference Schemes for Hyperbolic Equations with High Order of Accuracy,” Comm. Pure Appl. Math., 1964, 17, 381-398; see also Roe, P. L., “Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes,” J. Comput. Phys., 1981, 43, 357-372.
There are numerous approaches to performing reservoir simulation. Some of them, pertinent to problems in the oil and gas industry are described in U.S. Pat. No. 6,662,146. The patent describes how to perform reservoir simulation by solving a mixed implicit-IMPES matrix equation.
There is an approach to predicting behavior of a subterranean formation, described in U.S. Pat. No. 6,052,520. The patent describes how to estimate saturations from linearized relative permeabilities and capillary pressures.
There is an approach to performing oilfield simulation operations, described in U.S. Patent Application Publication No. 2009/0055141. The publication describes how to determine a time-step for simulating a reservoir using a reservoir model, calculating a plurality of conditions of the reservoir model corresponding to the time-step, simulating a cell of the reservoir model using an IMPES system to obtain a first simulation result, using the reservoir model with a fully implicit method (FIM) system to obtain a second simulation result, and performing an oilfield operation based on the first and second simulation results.
There is an approach described for a new IMPSAT model in J. Haukas, I. Aavatsmark, and M. Espedal, U. of Bergen, and E. Reiso, “A Comparison of Two Different IMPSAT Models in Compositional Simulation,” SPE J. 148-149 (2007). Isochoric variables are solved explicitly, and compared to a conventional IMPSAT model.
There is an approach for IMPES stability, described in K. H. Coats, “IMPES Stability: Selection of Stable Timesteps,” SPE J., 5-6 (2003). Stability criterion are used to set time steps, or as a switching criterion in an adaptive implicit model.
There is an approach for reservoir simulation described in Lu, B. and Alshallan, T., “Iteratively Coupled Reservoir Simulation for Multiphase Flow,” paper SPE 110114 presented at the SPE Annual Technical Conference and Exhibition, Anaheim, Calif., Nov. 11-14, 2007. Using the iterative approach allegedly reduces computational time by 30-40% over other approaches using the IMPES and fully implicit methods (FIM).
There is also an approach describing stability criteria in Wan, J. “General Stability Criteria for Compositional and Black-Oil Models,” paper 93096-MS presented at the SPE Reservoir Simulation Symposium, The Woodlands, Tex., Jan. 31-Feb. 2, 2005. General stability criteria are derived with various levels of implicitness. The derived criteria may provide a reasonable measure in the most general models.